## M1 Matrices: Introduction

• A matrix is a rectangular array of elements.

• Matrices are usually denoted by upper case letters.

• The elements are usually written within brackets.

• The order or shape of the matrix is determined by the number of rows and columns of the matrix.

• The number of rows is always given first then the number of columns. Example. \begin{align*} A & =\left[\begin{array}{ccc} 1 & 2 & -9\\ 2 & 5 & -3 \end{array}\right] \end{align*}

$$A$$ has 2 rows and 3 columns and is called a $$2\times3$$ matrix.1 This is verbally stated as a 2 by 3 matrix.

A matrix with $$m$$ rows and $$n$$ columns is called a matrix of order $$m\times n$$.2 This is verbally termed an “m by n matrix”.

### Square Matrix

A matrix with the same number of rows and columns is called a square matrix.

Example: \begin{align*} B & =\left[\begin{array}{cc} 2 & 3\\ 2 & 5 \end{array}\right] \end{align*}

$$B$$ is a square $$2\times2$$ matrix

### Unit Matrix

A unit (or identity) matrix is a square matrix with diagonal elements equal to one, and all other elements equal to zero. The unit matrix is usually denoted by $$I$$.

$$I_{3}$$ is a $$3\times3$$ unit matrix

Example: \begin{align*} I_{3} & =\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right] \end{align*}

### Row Matrix

A matrix with one row is called a row matrix.

Example: \begin{align*} D & =\left[\begin{array}{cccc} 2 & 1 & 0 & 4\end{array}\right] \end{align*} is a $$1\times4$$ row matrix

### Column Matrix

A matrix with one column is called a column matrix.

Example: \begin{align*} E & =\left[\begin{array}{c} 2\\ -4\\ 1 \end{array}\right] \end{align*}

is a $$3\times1$$ column matrix

### Zero Matrix

A zero matrix has all elements equal to zero. A zero matrix can be written as $$0$$.

Example: \begin{align*} 0 & =\left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right] \end{align*}

is a $$2\times2$$ zero matrix

### Equal Matrices

For two matrices to be equal, they must be the same shape and the corresponding elements must be equal.

If $$A$$ equals $$B$$ then \begin{align*} A & =\left[\begin{array}{ccc} 2 & 5 & b\\ 5 & 3 & 1\\ 2 & 0 & -2 \end{array}\right] & B & =\left[\begin{array}{ccc} 2 & 5 & 7\\ 5 & a & 1\\ 2 & 0 & -2 \end{array}\right] \end{align*}

If $$A$$ and $$B$$ are equal then $$a$$ = 3 and $$b$$ = 7.

### Exercise:

1. Write down the order of the following matrices.
1. $$\left[\begin{array}{ccc} 7 & -5 & 0\\ 6 & 2 & -1 \end{array}\right]$$

2. $$\left[\begin{array}{cc} 0 & 2\\ 1 & 1 \end{array}\right]$$

3. $$\left[\begin{array}{c} 2\\ -4\\ 1\\ 1 \end{array}\right]$$

4. $$\left[\begin{array}{cc} 1 & 1\\ 3 & 0\\ -2 & 3 \end{array}\right]$$

1. Which of the following matrices are equal?

\begin{align*} A & =\left[\begin{array}{cc} 3 & 0\\ 1 & -2 \end{array}\right] & B & =\left[\begin{array}{cc} 3 & 1\end{array}\right] & C & =\left[\begin{array}{cc} 3 & 0\end{array}\right] & D & =\left[\begin{array}{cc} 3 & 0\\ 1 & -2 \end{array}\right] \end{align*}

\begin{align*} E & =\left[\begin{array}{ccc} 3 & 5 & 1\\ 2 & 0 & 1 \end{array}\right] & F & =\left[\begin{array}{cc} 0 & 3\end{array}\right] & G & =\left[\begin{array}{ccc} 3 & 5 & 1\\ 2 & 0 & 1\\ 1 & 3 & 0 \end{array}\right] \end{align*}

$\begin{array}{lllllllccc} 1.\;a)\,2\times3 & & b)\,2\times2 & & c)\,4\times1 & & d)\,3\times2 & & & 2.\,A\textrm{ and D }\end{array}$